Vedic mathematics is an ancient system of mathematics discovered in the Vedas. It uses unique calculation techniques based on 16 sutras or formulae. These sutras allow mathematical problems in arithmetic, algebra, geometry and trigonometry to be solved very quickly mentally. Some key sutras include Ekadhikena Purvena for squaring numbers ending in 5, Nikhilam Navatashcaramam Dasatah for multiplication near multiples of 10, and Urdhva Tiryagbhyam for general multiplication and division of large numbers.

1. VEDICVEDIC
MATHEMATICSMATHEMATICS

2. What is Vedic Mathematics ?
 Vedic mathematics is the name
given to the ancient system of
mathematics which was
rediscovered from the Vedas.
 It’s a unique technique of
calculations based on simple
principles and rules , with which
any mathematical problem - be it
arithmetic, algebra, geometry or
trigonometry can be solved
mentally.

3. Why Vedic Mathematics?Why Vedic Mathematics?
 It helps a person to solve problems 10-15 times faster.
 It reduces burden (Need to learn tables up to nine only)
 It provides one line answer.
 It is a magical tool to reduce scratch work and finger
counting.
 It increases concentration.
 Time saved can be used to answer more questions.
 Improves concentration.
 Logical thinking process gets enhanced.

4. Base of Vedic MathematicsBase of Vedic Mathematics
 Vedic
Mathematics now
refers to a set of
sixteen
mathematical
formulae or sutras
and their
corollaries derived
from the Vedas.

5. Base of Vedic MathematicsBase of Vedic Mathematics
Vedic
Mathematics now
refers to a set of
sixteen
mathematical
formulae or sutras
and their
corollaries derived
from the Vedas.

6. EKĀDHIKENA PŪRVEŅAEKĀDHIKENA PŪRVEŅA
The Sutra
(formula)
Ekādhikena
Pūrvena means:
“By one more than
the previous one”.
 This Sutra is
used to the
‘Squaring of
numbers ending
in 5’.

7. ‘Squaring of numbers ending
in 5’.
Conventional Method
65 X 65
6 5
X 6 5
3 2 5
3 9 0 X
4 2 2 5
Vedic Method
65 X 65 = 4225
( 'multiply the
previous digit 6 by
one more than
itself 7. Than write
25 )

8. NIKHILAM
NAVATAS’CHARAMAM
DASATAH
The Sutra (formula)
NIKHILAM
NAVATAS’CHARA
MAM DASATAH
means :
“all from 9 and the
last from 10”
This formula can
be very effectively
applied in
multiplication of
numbers, which are
nearer to bases like
10, 100, 1000 i.e., to
the powers of 10
(eg: 96 x 98 or 102
x 104).

9. Case I :
When both the numbers are
lower than the base.
 Conventional Method
97 X 94
9 7
X 9 4
3 8 8
8 7 3 X
9 1 1 8
 Vedic Method
9797 33
XX 9494 66
9 1 1 89 1 1 8

10. Case ( ii) : When both theCase ( ii) : When both the
numbers are higher than thenumbers are higher than the
basebase
 Conventional
Method
103 X 105
103
X 105
5 1 5
0 0 0 X
1 0 3 X X
1 0, 8 1 5
 Vedic Method
For Example103 X 105For Example103 X 105
103103 33
XX 105 55
1 0, 8 1 5

11. Case III: When one number isCase III: When one number is
more and the other is lessmore and the other is less
than the base.than the base.
 Conventional Method
103 X 98
103
X 98
8 2 4
9 2 7 X
1 0, 0 9 4
 Vedic Method
103103 33
XX 98 -2
1 0, 0 9 4

12. ĀNURŨPYENA
The Sutra (formula)
ĀNURŨPYENA
means :
'proportionality '
or
'similarly '
This Sutra is highlyThis Sutra is highly
useful to finduseful to find
products of twoproducts of two
numbers whennumbers when
both of them areboth of them are
near the Commonnear the Common
bases like 50, 60,bases like 50, 60,
200 etc (multiples200 etc (multiples
of powers of 10).of powers of 10).

13. ĀNURŨPYENA
Conventional Method
46 X 43
4 6
X 4 3
1 3 8
1 8 4 X
1 9 7 8
Vedic Method
4646 -4-4
XX 43 -7-7
1 9 7 8

14. ĀNURŨPYENA
Conventional Method
58 X 48
5 8
X 4 8
4 6 4
2 4 2 X
2 8 8 4
Vedic Method
5858 88
XX 4848 -2-2
2 82 8 8 4

15. URDHVA TIRYAGBHYAM
The Sutra (formula)
URDHVA
TIRYAGBHYAM
means :
““Vertically and crossVertically and cross
wise”wise”
This the generalThis the general
formula applicableformula applicable
to all cases ofto all cases of
multiplication andmultiplication and
also in the divisionalso in the division
of a large numberof a large number
by another largeby another large
number.number.

16. Two digit multiplication byby
URDHVA TIRYAGBHYAM
The Sutra (formula)
URDHVA
TIRYAGBHYAM
means :
““Vertically and crossVertically and cross
wise”wise”
 Step 1Step 1: 5×2=10, write: 5×2=10, write
down 0 and carry 1down 0 and carry 1
 Step 2Step 2: 7×2 + 5×3 =: 7×2 + 5×3 =
14+15=29, add to it14+15=29, add to it
previous carry overprevious carry over
value 1, so we have 30,value 1, so we have 30,
now write down 0 andnow write down 0 and
carry 3carry 3
 Step 3Step 3: 7×3=21, add: 7×3=21, add
previous carry overprevious carry over
value of 3 to get 24,value of 3 to get 24,
write it down.write it down.
 So we have 2400 as theSo we have 2400 as the
answer.answer.

17. Two digit multiplication byby
URDHVA TIRYAGBHYAM
Vedic Method
4 6
X 4 3
1 9 7 8

18. Three digit multiplication by
URDHVA TIRYAGBHYAM
Vedic Method
103
X 105
1 0, 8 1 5

19. YAVDUNAM
TAAVDUNIKRITYA
VARGANCHA YOJAYET
This sutra means
whatever the extent
of its deficiency,
lessen it still
further to that very
extent; and also set
up the square of
that deficiency.
This sutra is very
handy in
calculating squares
of numbers
near(lesser) to
powers of 10

20. YAVDUNAM
TAAVDUNIKRITYA
VARGANCHA YOJAYET
98
2
= 9604
 The nearest power of 10 to 98 is 100.The nearest power of 10 to 98 is 100.
Therefore, let us take 100 as our base.Therefore, let us take 100 as our base.
 Since 98 is 2 less than 100, we call 2 asSince 98 is 2 less than 100, we call 2 as
the deficiency.the deficiency.
 Decrease the given number further by anDecrease the given number further by an
amount equal to the deficiency. i.e.,amount equal to the deficiency. i.e.,
perform ( 98 -2 ) = 96. This is the left sideperform ( 98 -2 ) = 96. This is the left side
of our answer!!.of our answer!!.
 On the right hand side put the square ofOn the right hand side put the square of
the deficiency, that is square of 2 = 04.the deficiency, that is square of 2 = 04.
 Append the results from step 4 and 5 toAppend the results from step 4 and 5 to
get the result. Hence the answer is 9604.get the result. Hence the answer is 9604.
NoteNote :: While calculating step 5, the number of digits in the squared number (04)While calculating step 5, the number of digits in the squared number (04)
should be equal to number of zeroes in the base(100).should be equal to number of zeroes in the base(100).

21. YAVDUNAM
TAAVDUNIKRITYA
VARGANCHA YOJAYET
103
2
= 10609
 The nearest power of 10 to 103 is 100.The nearest power of 10 to 103 is 100.
Therefore, let us take 100 as our base.Therefore, let us take 100 as our base.
 Since 103 is 3 more than 100 (base), weSince 103 is 3 more than 100 (base), we
call 3 as the surplus.call 3 as the surplus.
 Increase the given number further by anIncrease the given number further by an
amount equal to the surplus. i.e., performamount equal to the surplus. i.e., perform
( 103 + 3 ) = 106. This is the left side of( 103 + 3 ) = 106. This is the left side of
our answer!!.our answer!!.
 On the right hand side put the square ofOn the right hand side put the square of
the surplus, that is square of 3 = 09.the surplus, that is square of 3 = 09.
 Append the results from step 4 and 5 toAppend the results from step 4 and 5 to
get the result.Hence the answer is 10609.get the result.Hence the answer is 10609.
NoteNote :: while calculating step 5, the number of digits in the squared number (09)while calculating step 5, the number of digits in the squared number (09)
should be equal to number of zeroes in the base(100).should be equal to number of zeroes in the base(100).

22. YAVDUNAM
TAAVDUNIKRITYA
VARGANCHA YOJAYET
1009
2
= 1018081

23. SAŃKALANA –
VYAVAKALANĀBHYAM
The Sutra (formula)
SAŃKALANA –
VYAVAKALANĀ
BHYAM
means :
'by addition and by'by addition and by
subtraction'subtraction'
It can be applied inIt can be applied in
solving a specialsolving a special
type of simultaneoustype of simultaneous
equations where theequations where the
x - coefficients andx - coefficients and
the y - coefficientsthe y - coefficients
are foundare found
interchanged.interchanged.

24. SAŃKALANA –
VYAVAKALANĀBHYAM
Example 1:
45x – 23y = 113
23x – 45y = 91
 Firstly add them,Firstly add them,
( 45x – 23y ) + ( 23x – 45y ) = 113 + 91( 45x – 23y ) + ( 23x – 45y ) = 113 + 91
68x – 68y = 204 68x – 68y = 204
x – y = 3x – y = 3
 Subtract one from other,Subtract one from other,
( 45x – 23y ) – ( 23x – 45y ) = 113 – 91( 45x – 23y ) – ( 23x – 45y ) = 113 – 91
22x + 22y = 2222x + 22y = 22
x + y = 1x + y = 1
 Rrepeat the same sutra,Rrepeat the same sutra,
we getwe get x = 2x = 2 andand y = - 1y = - 1

25. SAŃKALANA –
VYAVAKALANĀBHYAM
Example 2:
1955x – 476y = 2482
476x – 1955y = - 4913
Just add,Just add,
2431( x – y ) = - 24312431( x – y ) = - 2431
x – y = -1x – y = -1
 Subtract,Subtract,
1479 ( x + y ) = 73951479 ( x + y ) = 7395
x + y = 5x + y = 5
Once again add,Once again add,
2x = 42x = 4 x = 2x = 2
subtractsubtract
- 2y = - 6- 2y = - 6 y = 3y = 3

26. ANTYAYOR DAŚAKE'PI
The Sutra (formula)
ANTYAYOR
DAŚAKE'PI
means :
‘‘ Numbers of whichNumbers of which
the last digitsthe last digits
added up giveadded up give
10.’10.’
 This sutra is helpful inThis sutra is helpful in
multiplying numbers whosemultiplying numbers whose
last digits add up to 10(orlast digits add up to 10(or
powers of 10). The remainingpowers of 10). The remaining
digits of the numbers shoulddigits of the numbers should
be identical.be identical.
For ExampleFor Example: In multiplication: In multiplication
of numbersof numbers
 25 and 25,25 and 25,
2 is common and 5 + 5 = 102 is common and 5 + 5 = 10
 47 and 43,47 and 43,
4 is common and 7 + 3 = 104 is common and 7 + 3 = 10
 62 and 68,62 and 68,
 116 and 114.116 and 114.
 425 and 475425 and 475

27. ANTYAYOR DAŚAKE'PI
Vedic Method
6 7
X 6 3
4 2 2 1
 The same rule worksThe same rule works
when the sum of the lastwhen the sum of the last
2, last 3, last 4 - - - digits2, last 3, last 4 - - - digits
added respectively equaladded respectively equal
to 100, 1000, 10000 -- - - .to 100, 1000, 10000 -- - - .
 The simple point toThe simple point to
remember is to multiplyremember is to multiply
each product by 10, 100,each product by 10, 100,
1000, - - as the case may1000, - - as the case may
be .be .
 You can observe that thisYou can observe that this
is more convenient whileis more convenient while
working with the productworking with the product
of 3 digit numbersof 3 digit numbers

28. ANTYAYOR DAŚAKE'PI
892 X 808
= 720736
Try Yourself :Try Yourself :
A)A) 398 X 302398 X 302
= 120196= 120196
B)B) 795 X 705795 X 705
= 560475= 560475

29. LOPANA
STHÂPANÂBHYÂM
The Sutra (formula)
LOPANA
STHÂPANÂBHYÂ
M
means :
'by alternate'by alternate
elimination andelimination and
retention'retention'
 Consider the case ofConsider the case of
factorization of quadraticfactorization of quadratic
equation of typeequation of type
axax22
+ by+ by22
+ cz+ cz22
+ dxy + eyz + fzx+ dxy + eyz + fzx
 This is a homogeneousThis is a homogeneous
equation of second degreeequation of second degree
in three variables x, y, z.in three variables x, y, z.
 The sub-sutra removesThe sub-sutra removes
the difficulty and makesthe difficulty and makes
the factorization simple.the factorization simple.

30. LOPANA
STHÂPANÂBHYÂM
Example :
3x 2
+ 7xy + 2y 2
+ 11xz + 7yz + 6z 2
 Eliminate z and retain x, y ;
factorize
3x 2
+ 7xy + 2y 2
= (3x + y) (x + 2y)
 Eliminate y and retain x, z;
factorize
3x 2
+ 11xz + 6z 2
= (3x + 2z) (x + 3z)
 Fill the gaps, the given expression
(3x + y + 2z) (x + 2y + 3z)
 Eliminate z by putting z = 0Eliminate z by putting z = 0
and retain x and y andand retain x and y and
factorize thus obtained afactorize thus obtained a
quadratic in x and y byquadratic in x and y by
means ofmeans of AdyamadyenaAdyamadyena
sutra.sutra.
 Similarly eliminate y andSimilarly eliminate y and
retain x and z and factorizeretain x and z and factorize
the quadratic in x and z.the quadratic in x and z.
 With these two sets ofWith these two sets of
factors, fill in the gapsfactors, fill in the gaps
caused by the eliminationcaused by the elimination
process of z and yprocess of z and y
respectively. This givesrespectively. This gives
actual factors of theactual factors of the

31. GUNÌTA SAMUCCAYAH -
SAMUCCAYA GUNÌTAH
Example :
3x 2
+ 7xy + 2y 2
+ 11xz + 7yz + 6z 2
 Eliminate z and retain x, y ;
factorize
3x 2
+ 7xy + 2y 2
= (3x + y) (x + 2y)
 Eliminate y and retain x, z;
factorize
3x 2
+ 11xz + 6z 2
= (3x + 2z) (x + 3z)
 Fill the gaps, the given expression
(3x + y + 2z) (x + 2y + 3z)
 Eliminate z by putting z = 0Eliminate z by putting z = 0
and retain x and y andand retain x and y and
factorize thus obtained afactorize thus obtained a
quadratic in x and y byquadratic in x and y by
means ofmeans of AdyamadyenaAdyamadyena
sutra.sutra.
 Similarly eliminate y andSimilarly eliminate y and
retain x and z and factorizeretain x and z and factorize
the quadratic in x and z.the quadratic in x and z.
 With these two sets ofWith these two sets of
factors, fill in the gapsfactors, fill in the gaps
caused by the eliminationcaused by the elimination
process of z and yprocess of z and y
respectively. This givesrespectively. This gives
actual factors of theactual factors of the

32. Prepared By:
Nitin
Chhaperwal